// Numbas version: exam_results_page_options {"name": "Luis's copy of Integration by parts", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"ungrouped_variables": ["a", "c", "b", "s3", "s2", "s1", "a1", "a2"], "name": "Luis's copy of Integration by parts", "variablesTest": {"condition": "", "maxRuns": 100}, "preamble": {"js": "", "css": ""}, "parts": [{"prompt": "\n\t\t\t
$I=\\displaystyle \\int \\simplify[std]{({a}x+{b})*e^({c}x)} dx $
You are given that the answer is of the form \\[I=g(x)e^{\\var{c}x}+C\\] for a polynomial $g(x)$. You have to find $g(x)$.
$g(x)=\\;$[[0]]
\n\t\t\tInput all numbers as fractions or integers and not decimals.
\n\t\t\tYou can get help by clicking on Show steps. You will lose 1 mark if you do.
\n\t\t\t", "gaps": [{"checkvariablenames": false, "showpreview": true, "vsetrangepoints": 5, "answersimplification": "all", "type": "jme", "expectedvariablenames": [], "scripts": {}, "marks": 2, "checkingaccuracy": 0.001, "checkingtype": "absdiff", "answer": "({a}/{c})*x+{c*b-a}/{c^2}", "showCorrectAnswer": true, "vsetrange": [0, 1], "notallowed": {"message": "Do not input numbers as decimals, only as integers without the decimal point, or fractions
", "strings": ["."], "showStrings": false, "partialCredit": 0}}], "steps": [{"type": "information", "prompt": "\n\t\t\t\t\t \n\t\t\t\t\t \n\t\t\t\t\tThe formula for integrating by parts is
\\[ \\int u\\frac{dv}{dx} dx = uv - \\int v \\frac{du}{dx} dx. \\]
Use the result from the first part to find:
\n\t\t\t$\\displaystyle I=\\int \\simplify[std]{({a}x+{b})^2*e^({c}x)} dx $
\n\t\t\tYou are given that the answer is of the form \\[I=h(x)e^{\\var{c}x}+C\\] for a polynomial $h(x)$. You have to find $h(x)$.
\n\t\t\t$h(x)=\\;$[[0]]
\n\t\t\tInput all numbers as fractions or integers and not decimals.
\n\t\t\t", "gaps": [{"checkvariablenames": false, "showpreview": true, "vsetrangepoints": 5, "answersimplification": "all", "type": "jme", "expectedvariablenames": [], "scripts": {}, "marks": 3, "checkingaccuracy": 0.001, "checkingtype": "absdiff", "answer": "{a^2}/{c}*x^2+{2*a*b*c-2*a^2}/{c^2}*x+{b^2*c^2-2*a*b*c+2*a^2}/{c^3}", "showCorrectAnswer": true, "vsetrange": [0, 1], "notallowed": {"message": "Do not input numbers as decimals, only as integers without the decimal point, or fractions
", "strings": ["."], "showStrings": false, "partialCredit": 0}}], "showCorrectAnswer": true, "type": "gapfill", "scripts": {}, "marks": 0}], "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "type": "question", "functions": {}, "variables": {"s3": {"name": "s3", "templateType": "anything", "description": "", "definition": "random(1,-1)", "group": "Ungrouped variables"}, "a1": {"name": "a1", "templateType": "anything", "description": "", "definition": "s1*random(1..9)", "group": "Ungrouped variables"}, "s2": {"name": "s2", "templateType": "anything", "description": "", "definition": "random(1,-1)", "group": "Ungrouped variables"}, "s1": {"name": "s1", "templateType": "anything", "description": "", "definition": "random(1,-1)", "group": "Ungrouped variables"}, "b": {"name": "b", "templateType": "anything", "description": "", "definition": "s1*random(1..9)", "group": "Ungrouped variables"}, "a2": {"name": "a2", "templateType": "anything", "description": "", "definition": "s2*random(1..9)", "group": "Ungrouped variables"}, "c": {"name": "c", "templateType": "anything", "description": "", "definition": "s3*random(2..5)", "group": "Ungrouped variables"}, "a": {"name": "a", "templateType": "anything", "description": "", "definition": "random(2..5)", "group": "Ungrouped variables"}}, "variable_groups": [], "question_groups": [{"name": "", "pickQuestions": 0, "pickingStrategy": "all-ordered", "questions": []}], "showQuestionGroupNames": false, "tags": ["Calculus", "MAS1601", "Steps", "algebraic manipulation", "checked2015", "exponential function", "integration", "integration by parts", "integration of exponential function"], "advice": "\n\ta)
\n\tThe formula for integrating by parts is
\\[ \\int u\\frac{dv}{dx} dx = uv - \\int v \\frac{du}{dx} dx. \\]
We choose $u = \\simplify[std]{{a}x+{b}}$ and $\\displaystyle\\frac{dv}{dx} = \\simplify[std]{e^({c}x)}$.
\n\tSo $\\displaystyle \\frac{du}{dx} = \\var{a}$ and $\\displaystyle v = \\simplify[std]{(1/{c})*e^({c}*x)}$.
\n\tHence,
\\[ \\begin{eqnarray} \\int \\simplify[std]{({a}*x+{b})*e^({c}*x)} dx &=& uv - \\int v \\frac{du}{dx} dx \\\\ &=& \\simplify[std]{({1}/{c})*({a}x+{b})*e^({c}x) - (1/{c})*Int(({a})*e^({c}x),x)} \\\\ &=& \\simplify[std]{(1/{c})*({a}x+{b})*e^({c}x) -({a}/{c^2})*e^({c}x) + C}\\\\ &=& \\simplify[std]{(({a}x+{b})/{c}-{a}/{c^2})*e^({c}*x) + C}\\\\ &=& \\simplify[std]{(({a}/{c})x+{b*c-a}/{c^2})*e^({c}*x) + C} \\end{eqnarray} \\]
Hence $\\displaystyle \\simplify[std]{g(x)=({a}/{c})*x+{c*b-a}/{c^2}}$
\n\tb)
\n\tFor this part we choose $u = \\simplify[std]{({a}x+{b})^2}$ and $\\displaystyle \\frac{dv}{dx} = \\simplify[std]{e^({c}x)}$.
\n\tSo $\\displaystyle \\frac{du}{dx}$ = $\\simplify[std]{{2*a}*({a}*(x)+{b})}$ and $\\displaystyle v = \\simplify[std]{(1/{c})*e^({c}*x)}$.
\n\tHence,
\\[ \\begin{eqnarray*}I= \\int \\simplify[std]{({a}*x+{b})^2*e^({c}*x)} dx &=& uv - \\int v \\frac{du}{dx} dx \\\\ &=& \\simplify[std]{({1}/{c})*({a}x+{b})^2*e^({c}x) - (1/{c})*Int({2*a}*({a}x+{b})*e^({c}x),x)} \\\\ &=& \\simplify[std]{(1/{c})*({a}x+{b})^2*e^({c}x) -({2*a}/{c})*Int(({a}x+{b})*e^({c}x),x)}\\dots (*) \\end{eqnarray*}\\]
But in Part a we have aready worked out $\\displaystyle \\simplify[std]{Int(({a}x+{b})*e^({c}*x),x)=(({a}/{c})*x+({c*b-a}/{c^2}))*e^({c}*x)+C}$
\n\tSo on substituting this in equation (*) we find:
\\[ \\begin{eqnarray*}I&=& \\simplify[std]{(1/{c})*({a}x+{b})^2*e^({c}x) -({2*a}/{c})*(({a}/{c})*x+({c*b-a}/{c^2}))*e^({c}*x)+C}\\\\ &=& \\simplify[std]{({a^2}/{c}*x^2+{2*a*b*c-2*a^2}/{c^2}*x+{b^2*c^2-2*a*b*c+2*a^2}/{c^3})*e^({c}x) +C} \\end{eqnarray*}\\]
Hence $\\displaystyle \\simplify[std]{h(x)={a^2}/{c}*x^2+{2*a*b*c-2*a^2}/{c^2}*x+{b^2*c^2-2*a*b*c+2*a^2}/{c^3}}$
\n\t", "statement": "\n\tFind the following indefinite integrals.
\n\tInput all numbers as fractions or integers and not decimals.
\n\t", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "notes": "\n\t\t3/08/2012:
\n\t\tAdded tags.
\n\t\tAdded description.
\n\t\tChecked calculation. OK.
\n\t\tGot rid of redundant instructions about inputting constant of integration.
\n\t\tPenalised use of steps in first part, 1 mark. Added message to that effect in first part.
\n\t\tAdded message about not inputting decimals in appropriate places.
\n\t\tChanged marks reflecting the use of steps and degree of difficulty in second part.
\n\t\tImproved Advice display.
\n\t\t", "description": "Given $\\displaystyle \\int (ax+b)e^{cx}\\;dx =g(x)e^{cx}+C$, find $g(x)$. Find $h(x)$, $\\displaystyle \\int (ax+b)^2e^{cx}\\;dx =h(x)e^{cx}+C$.
"}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Luis Hernandez", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2870/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Luis Hernandez", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2870/"}]}